Higher set theory and mathematical practice

In a previous paper, I proved the consistency of a non-finitary system of logic based on the theory of types, which was shown to contain the axiom of reducibility in a form which seemed not to interfere with the classical construction of real numbers. A form of the system containing a strong axiom of choice was also proved consistent.It seems to me now that the real-number approach used in that paper, though valid, was not the most fruitful one. We can, on the lines therein suggested, prove the consistency of axioms closely resembling Tarski's twenty axioms for the real numbers; but this, from the standpoint of mathematical practice, is a pitifully small fragment of analysis. The consistency of a fairly strong set-theory can be proved, using the results of my previous paper, with little more difficulty than that of the Tarski axioms; this being the case, it would seem a saving in effort to derive the consistency of such a theory first, then to strengthen that theory (if possible) in such ways as can be shown to preserve consistency; and finally to derive from the system thus strengthened, if need be, a more usable real-number theory. The present paper is meant to achieve the first part of this program. The paragraphs of this paper are numbered consecutively with those of my previous paper, of which it is to be regarded as a continuation.

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Philosophy of Mathematics

Philosophy ◽

10.1093/obo/9780195396577-0069 ◽

2010 ◽

Author(s):

Otávio Bueno

Keyword(s):

Philosophy Of Science ◽

Set Theory ◽

Category Theory ◽

Mathematical Reasoning ◽

Philosophy Of Mathematics ◽

Mathematical Practice ◽

Mathematical Explanation ◽

Excellent Work ◽

Novel Approaches

Philosophy of mathematics is arguably one of the oldest branches of philosophy, and one that bears significant connections with core philosophical areas, particularly metaphysics, epistemology, and (more recently) the philosophy of science. This entry focuses on contemporary developments, which have yielded novel approaches (such as new forms of Platonism and nominalism, structuralism, neo-Fregeanism, empiricism, and naturalism) as well as several new issues (such as the significance of the application of mathematics, the role of visualization in mathematical reasoning, particular attention to mathematical practice and to the nature of mathematical explanation). Excellent work has also been done on particular philosophical issues that arise in the context of specific branches of mathematics, such as algebra, analysis, and geometry, as well as particular mathematical theories, such as set theory and category theory. Due to limitations of space, this work goes beyond the scope of the present entry.

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Three Roles of Empirical Information in Philosophy: Intuitions on Mathematics do Not Come for Free

KRITERION – Journal of Philosophy ◽

10.1515/krt-2021-0025 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Deborah Kant ◽

José Antonio Pérez-Escobar ◽

Deniz Sarikaya

Keyword(s):

Set Theory ◽

Philosophy Of Mind ◽

Experimental Philosophy ◽

Philosophy Of Mathematics ◽

Mathematical Practice ◽

The Other ◽

Empirical Information ◽

Philosophy Of Mathematical Practice ◽

Armchair Philosophy

Abstract This work gives a new argument for ‘Empirical Philosophy of Mathematical Practice’. It analyses different modalities on how empirical information can influence philosophical endeavours. We evoke the classical dichotomy between “armchair” philosophy and empirical/experimental philosophy, and claim that the latter should in turn be subdivided in three distinct styles: Apostate speculator, Informed analyst, and Freeway explorer. This is a shift of focus from the source of the information towards its use by philosophers. We present several examples from philosophy of mind/science and ethics on one side and a case study from philosophy of mathematics on the other. We argue that empirically informed philosophy of mathematics is different from the rest in a way that encourages a Freeway explorer approach, because intuitions about mathematical objects are often unavailable for non-mathematicians (since they are sometimes hard to grasp even for mathematicians). This consideration is supported by a case study in set theory.

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Constructive set theory

Journal of Symbolic Logic ◽

10.2307/2272159 ◽

1975 ◽

Vol 40 (3) ◽

pp. 347-382 ◽

Cited By ~ 99

Author(s):

John Myhill

Keyword(s):

Set Theory ◽

Mathematical Practice ◽

Partial Function ◽

Advanced Mathematics ◽

The Third ◽

Formal Systems ◽

Constructive Set Theory ◽

Definition Of

This paper is the third in a series collectively entitled Formal systems of intuitionistic analysis. The first two are [4] and [5] in the bibliography; in them I attempted to codify Brouwer's mathematical practice. In the present paper, which is independent of [4] and [5], I shall do the same for Bishop's book [1]. There is a widespread current impression, due partly to Bishop himself (see [2]) and partly to Goodman and the author (see [3]) that the theory of Gödel functionals, with quantifiers and choice, is the appropriate formalism for [1]. That this is not so is seen as soon as one really tries to formalize the mathematics of [1] in detail. Even so simple a matter as the definition of the partial function 1/x on the nonzero reals is quite a headache, unless one is prepared either to distinguish nonzero reals from reals (a nonzero real being a pair consisting of a real x and an integer n with ∣x∣ > 1/n) or, to take the Dialectica interpretation seriously, by adjoining to the Gödel system an axiom saying that every formula is equivalent to its Dialectica interpretation. (See [1, p. 19], [2, pp. 57–60] respectively for these two methods.) In more advanced mathematics the complexities become intolerable.

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On the strength of nonstandard analysis

Journal of Symbolic Logic ◽

10.1017/s0022481200031248 ◽

1986 ◽

Vol 51 (2) ◽

pp. 377-386 ◽

Cited By ~ 13

Author(s):

C. Ward Henson ◽

H. Jerome Keisler

Keyword(s):

Set Theory ◽

Nonstandard Analysis ◽

Mathematical Practice ◽

Second Order ◽

Erroneous Conclusion ◽

Second Order Arithmetic ◽

Correct Statement ◽

Classical Mathematics ◽

Order Arithmetic ◽

Almost All

It is often asserted in the literature that any theorem which can be proved using nonstandard analysis can also be proved without it. The purpose of this paper is to show that this assertion is wrong, and in fact there are theorems which can be proved with nonstandard analysis but cannot be proved without it. There is currently a great deal of confusion among mathematicians because the above assertion can be interpreted in two different ways. First, there is the following correct statement: any theorem which can be proved using nonstandard analysis can be proved in Zermelo-Fraenkel set theory with choice, ZFC, and thus is acceptable by contemporary standards as a theorem in mathematics. Second, there is the erroneous conclusion drawn by skeptics: any theorem which can be proved using nonstandard analysis can be proved without it, and thus there is no need for nonstandard analysis.The reason for this confusion is that the set of principles which are accepted by current mathematics, namely ZFC, is much stronger than the set of principles which are actually used in mathematical practice. It has been observed (see [F] and [S]) that almost all results in classical mathematics use methods available in second order arithmetic with appropriate comprehension and choice axiom schemes.

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